# 1.5 Factoring polynomials  (Page 3/6)

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Factor a. $\text{\hspace{0.17em}}2{x}^{2}+9x+9\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}6{x}^{2}+x-1$

a. $\text{\hspace{0.17em}}\left(2x+3\right)\left(x+3\right)\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}\left(3x-1\right)\left(2x+1\right)$

## Factoring a perfect square trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

$\begin{array}{ccc}\hfill {a}^{2}+2ab+{b}^{2}& =& {\left(a+b\right)}^{2}\hfill \\ & \text{and}& \\ \hfill {a}^{2}-2ab+{b}^{2}& =& {\left(a-b\right)}^{2}\hfill \end{array}$

We can use this equation to factor any perfect square trinomial.

## Perfect square trinomials

A perfect square trinomial can be written as the square of a binomial:

${a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}$

Given a perfect square trinomial, factor it into the square of a binomial.

1. Confirm that the first and last term are perfect squares.
2. Confirm that the middle term is twice the product of $\text{\hspace{0.17em}}ab.$
3. Write the factored form as $\text{\hspace{0.17em}}{\left(a+b\right)}^{2}.$

## Factoring a perfect square trinomial

Factor $\text{\hspace{0.17em}}25{x}^{2}+20x+4.$

Notice that $\text{\hspace{0.17em}}25{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ are perfect squares because $\text{\hspace{0.17em}}25{x}^{2}={\left(5x\right)}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}4={2}^{2}.\text{\hspace{0.17em}}$ Then check to see if the middle term is twice the product of $\text{\hspace{0.17em}}5x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ The middle term is, indeed, twice the product: $\text{\hspace{0.17em}}2\left(5x\right)\left(2\right)=20x.\text{\hspace{0.17em}}$ Therefore, the trinomial is a perfect square trinomial and can be written as $\text{\hspace{0.17em}}{\left(5x+2\right)}^{2}.$

Factor $\text{\hspace{0.17em}}49{x}^{2}-14x+1.$

${\left(7x-1\right)}^{2}$

## Factoring a difference of squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

${a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)$

We can use this equation to factor any differences of squares.

## Differences of squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

${a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)$

Given a difference of squares, factor it into binomials.

1. Confirm that the first and last term are perfect squares.
2. Write the factored form as $\text{\hspace{0.17em}}\left(a+b\right)\left(a-b\right).$

## Factoring a difference of squares

Factor $\text{\hspace{0.17em}}9{x}^{2}-25.$

Notice that $\text{\hspace{0.17em}}9{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}25\text{\hspace{0.17em}}$ are perfect squares because $\text{\hspace{0.17em}}9{x}^{2}={\left(3x\right)}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}25={5}^{2}.\text{\hspace{0.17em}}$ The polynomial represents a difference of squares and can be rewritten as $\text{\hspace{0.17em}}\left(3x+5\right)\left(3x-5\right).$

Factor $\text{\hspace{0.17em}}81{y}^{2}-100.$

$\left(9y+10\right)\left(9y-10\right)$

Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

## Factoring the sum and difference of cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

${a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)$

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

${a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)$

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.

${x}^{3}-{2}^{3}=\left(x-2\right)\left({x}^{2}+2x+4\right)$

The sign of the first 2 is the same as the sign between $\text{\hspace{0.17em}}{x}^{3}-{2}^{3}.\text{\hspace{0.17em}}$ The sign of the $\text{\hspace{0.17em}}2x\text{\hspace{0.17em}}$ term is opposite the sign between $\text{\hspace{0.17em}}{x}^{3}-{2}^{3}.\text{\hspace{0.17em}}$ And the sign of the last term, 4, is always positive .

## Sum and difference of cubes

We can factor the sum of two cubes as

${a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)$

We can factor the difference of two cubes as

${a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)$

#### Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
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Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
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mantu
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Modress
-x=7
Modress
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siame
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deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
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Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
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Rubben
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More example of algebra and trigo
What is Indices
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Navin